Global Wave Forecasts Using Spacecraft Data 



- 90° < e. < ^^+90° 



The sum for 0-^ satisfying the above inequality is defined as 



s,(fi) = Ls(i,,e.) 



(5) 



(6) 



and represents the sum of all contributions to the spectrum for a given f j trav- 

 eling within 90° to the wind. If S^ (f j) > S„ (f j), there is no change in the values 

 of s ( f J , 6* . ) , but if S^ ( f i ) < S<„ ( f .), there will be growth during the next time 

 step. The growth of each frequency band f ^ is then given by the solution to the 

 ordinary differential equation 



Jt Sw(fi) = A(v) 



1- 



Sw(fi) 



Sco(fi). 



+ B(v, fi)S,(fi) 



Sw(fi) 



Sco(fi)J 



A(v) 1 



Sw(fi)' 



s»(fi) 



1/2 



+ B(v. i.) S^({.) 



(7) 



where 



A(v) = 1.4x 10"^ V- 



(8) 



and 



B(v, f) = 6.27 (v/c)2 e' 



17(c/v)^ 



2fAf 



(9) 



for those frequency bands where 2Af = 1/180 and where Eq. (7) has units of 

 feet Vhour and v and c are in knots (18). Equation (7) can be integrated for the ini- 

 tial value s^(f. ) = to yield s^(f .; t^) so that one can solve for the effective 

 time tp corresponding to s^(f . ) at the start of the time step. From this, 

 s^ ( f j; tg + At) for the end of the 2-hour time step can be found. The incre- 

 mental growth 



Sw(fi) = S,(fi; t,+ At) - S,(f.; t^) 



is then spread out in direction according to Eq. (2) with 0^ corresponding to 

 0* = 0. The increments are not allowed to exceed the values given by Eq. (3) 

 for each 0^, although the shape of F(co, 0, v) is not always preserved. 



It would be possible to treat each s^ (f . , 0^) separately as an initial-value 

 problem. As the wind direction changed, such a procedure would tend to produce 

 spectra in excess of Eq. (1), whereas this procedure avoids this, at least in part. 



A modification based on the work of Snyder and Cox (17) and Barnett (24) 

 for the term A(v) in Eq. (8) is under investigation. The Phillips term would then 

 be a function of both wind speed and frequency (but still not direction according to 



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